Let `x,y,z in (0, (pi)/(2))` are first three consecutive terms of an arithmatic progression such that `cos x + cos y + cos z =1 and sin x + sin y + sin z = (1)/(sqrt2),` then which of the following is/are correct ?

Options

A. `cot y= sqrt2`
B. `cos (x-y) =(sqrt3 - sqrt2)/(2 sqrt2)`
C. `tan 2y = (2 sqrt2)/(3 )`
D. ` sin (x-y) + sin (y-z ) =0`

Answer

To solve the problem, we need to find the values of \(x\), \(y\), and \(z\) given that they are the first three consecutive terms of an arithmetic progression (AP) and satisfy the equations:

1. \( \cos x + \cos y + \cos z = 1 \)
2. \( \sin x + \sin y + \sin z = \frac{1}{\sqrt{2}} \)

### Step 1: Express \(x\), \(y\), and \(z\) in terms of \(y\) and the common difference \(\theta\)

Since \(x\), \(y\), and \(z\) are in AP, we can express them as:
- \(x = y - \theta\)
- \(y = y\)
- \(z = y + \theta\)

### Step 2: Substitute \(x\) and \(z\) in the cosine equation

Substituting \(x\) and \(z\) into the cosine equation:
\[ \cos(y - \theta) + \cos(y) + \cos(y + \theta) = 1 \]

Using the cosine addition and subtraction formulas:
\[ \cos(y - \theta) = \cos y \cos \theta + \sin y \sin \theta \]
\[ \cos(y + \theta) = \cos y \cos \theta - \sin y \sin \theta \]

Thus, the equation becomes:
\[ (\cos y \cos \theta + \sin y \sin \theta) + \cos y + (\cos y \cos \theta - \sin y \sin \theta) = 1 \]
This simplifies to:
\[ 2 \cos y \cos \theta + \cos y = 1 \]
Factoring out \(\cos y\):
\[ \cos y (2 \cos \theta + 1) = 1 \quad \text{(Equation 1)} \]

### Step 3: Substitute \(x\) and \(z\) in the sine equation

Now, substituting into the sine equation:
\[ \sin(y - \theta) + \sin(y) + \sin(y + \theta) = \frac{1}{\sqrt{2}} \]

Using the sine addition and subtraction formulas:
\[ \sin(y - \theta) = \sin y \cos \theta - \cos y \sin \theta \]
\[ \sin(y + \theta) = \sin y \cos \theta + \cos y \sin \theta \]

Thus, the equation becomes:
\[ (\sin y \cos \theta - \cos y \sin \theta) + \sin y + (\sin y \cos \theta + \cos y \sin \theta) = \frac{1}{\sqrt{2}} \]
This simplifies to:
\[ 2 \sin y \cos \theta + \sin y = \frac{1}{\sqrt{2}} \]
Factoring out \(\sin y\):
\[ \sin y (2 \cos \theta + 1) = \frac{1}{\sqrt{2}} \quad \text{(Equation 2)} \]

### Step 4: Divide Equation 1 by Equation 2

Now, we can divide Equation 1 by Equation 2:
\[ \frac{\cos y (2 \cos \theta + 1)}{\sin y (2 \cos \theta + 1)} = \frac{1}{\frac{1}{\sqrt{2}}} \]
This simplifies to:
\[ \frac{\cos y}{\sin y} = \sqrt{2} \quad \Rightarrow \quad \cot y = \sqrt{2} \quad \Rightarrow \quad \tan y = \frac{1}{\sqrt{2}} \]

### Step 5: Find \(y\)

From \(\tan y = \frac{1}{\sqrt{2}}\), we can find \(y\):
\[ y = \frac{\pi}{4} \]

### Step 6: Find \(\cos y\) and \(\sin y\)

Now, we can find:
\[ \cos y = \frac{1}{\sqrt{2}}, \quad \sin y = \frac{1}{\sqrt{2}} \]

### Step 7: Substitute \(y\) back to find \(x\) and \(z\)

Now we can substitute \(y\) back into Equations 1 and 2 to find \(\theta\):
From Equation 1:
\[ \frac{1}{\sqrt{2}}(2 \cos \theta + 1) = 1 \quad \Rightarrow \quad 2 \cos \theta + 1 = \sqrt{2} \]
\[ 2 \cos \theta = \sqrt{2} - 1 \quad \Rightarrow \quad \cos \theta = \frac{\sqrt{2} - 1}{2} \]

### Step 8: Verify the conditions

Finally, we need to verify that the values satisfy the original conditions:
1. Check \( \cos x + \cos y + \cos z = 1 \)
2. Check \( \sin x + \sin y + \sin z = \frac{1}{\sqrt{2}} \)

### Conclusion

Based on the calculations, we find that:
- \(y = \frac{\pi}{4}\)
- \(x\) and \(z\) can be derived from \(y\) and \(\theta\).